
Let us remind the reader how the alt-dPI equation appeared for the first time. Going back to the scalar context, in Magnus' work [1]}, associated with the weight functions solution of the Pearson equation
\( W^{\prime } (z) = \big ( z^2 + t \big ) W (z)\) , we can find the following scalar alternate discrete Painlevé I  system
\(\gamma _n + \gamma _{n+1} + \beta _n^2 + t &= 0, \\n + \gamma _n \, \big ( \beta _n + \beta _{n-1} \big ) &= 0,\) 
